Assigning Judges to Competitions Using Tabu Search Approach

Transcription

1 Assigning Judges to Competitions Using Tabu Search Approach Amina Lamghari Jacques A. Ferland Computer science and OR dept. University of Montreal Faculty of Information Technology University of Science Vietnam National University of Ho Chi Minh City March 2010

2 Problem background The John Molson Business School International Case Competition Takes place every year for more than 30 years at Concordia University in Montreal 30 teams of business school students coming from top international universities Partitioned in 5 groups of 6 teams First part of the competition is a round-robin tournament including 5 rounds where each team competes against each of the other 5 teams of its group The three best teams move to the finals

3 Judge Assignment to the competitions of a round. Constraints Objective function

4 Penalty for competitions with 1 judge Maximize the number of competitions with 5 judges Number of judges with expertise k assigned to j a ij = 1 iff i admissible for j At least one lead judge 1, 33 or or 55 judges assigned

5 Metaheuristic Solution Approach Initial Solution First Stage Structured Neighborhood Tabu Search to reduce the number of competitions with 1 or 3 judges Second Stage Tabu Search to improve the diversity of the fields of expertise of the judges assigned to a competition Diversification strategy Adaptive memory of best solutions generated Crossover to generate a new initial solution

6 Initial Solution Two different processes i) Random Assign randomly 1 lead judge to each competition Hence all competitions have 1 judge assigned ii) HLA-HOA Constructive Heuristic 1. Assign 1 lead judge to each competition 2. Assign a first pair of additional judges to each competition 3. Assign a second pair of additional judges to as many competitions as possible Look ahead features making further assignments easier.

7 First Stage Structured Neighborhood Tabu Search Neighborhood Reassignment of a pair of judges (i, r) from competition j to competition l : x ( i, r, j, l) x ( i, r, j, l) in the neighborhood if the solution is feasible

8 Structured Neighborhood Reassignment of pair (i, r) from comp. j to comp. l improving improving or deteriorating deteriorating from j to l Impact on the objective function

9 Search Strategy improving Using sequentially V 1 (x), V 2 (x), V 3 (x). Exhaustive search from j to l Impact on the objective function

10 Search Strategy Using sequentially V 4 (x),, V 8 (x). improving or deteriorating deteriorating No exhaustive search. After any reassignment in V 4 (x), return to V 2 (x). After any reassignment in V k (x), k = 5,, 8, return to V 1 (x).

11 Search Strategy Similarity with Variable Neighborhood Search (VNS) But the search strategy strongly depend on the partition of the neighborhood, and the potential improvement associated with the different subsets improving improving or deteriorating deteriorating from j to l Impact on the objective function

12 Tabu list When x ( i, r, j, l) becomes the current solution (i, j) and (r, j) are included in the Tabu list Tabu solution x ( i, r, j, l) is Tabu if (i, l) and (r, l) are in the Tabu list Aspiration criterion x ( i, r, j, l) the best value reached so far satisfies the aspiration criterion if its value is better than

13 Selection of the new current solution in the neighborhood Two different strategies compared numerically Best Generate the entire neighborhood and select the best solution in it First During the generation of the neighborhood, select - the first non Tabu solution improving the value of the current solution or - the first solution satisfying the aspiration criterion

14 Stopping criteria - No improvement possible 5 judges assigned to each competition or 3 or 5 judges assigned to each competition and only 0 or 1 judge is not assigned - nitermax successive iterations where the objective function does not improve

15 Second Stage Tabu Search Neighborhood Exchange judge i of competition j and judge r of competition l x ( i, j, r, l) in the neighborhood if the solution is feasible Tabu list When x ( i, j, r, l) becomes the current solution (i, j) and (r, l) are included in the Tabu list Tabu solution x ( i, j, r, l) is Tabu if (i, l) and (r, j) are in the Tabu list

16 Aspiration criterion (idem) Selection of the new current solution in the neighborhood (idem) Stopping criteria - No improvement possible in each competition, all judges have different expertise or lower bound known a priori for the problem is reached - nitermax successive iterations where the objective function does not improve

17 Diversification Strategy Adaptive Memory Γ: set of best solutions found so far Uniform crossover x xbest selected randomly in Γ Uniform crossover of x and xbest to generate x 0 % of elements from xbest decreases with the number of recent successive major iterations where the objective does not improve

18 Diversification Strategy Adaptive Memory Γ: set of best solutions found so far Uniform crossover x xbest selected randomly in Γ Uniform crossover of x and xbest to generate x 0 % of elements from xbest decreases with the number of recent successive major iterations where the objective does not improve I( j, x 0 ) = I( j, xbest) I( j, x) if j M otherwise. best i: judge i is assigned to individual I( j, x) = competition j in solution x

19 Repair process Eliminate duplicate judges from x 0 Bias in favour of elements from xbest Look ahead feature to have a lead judge in each competition Assign a lead judge to those competitions missing one For competition having 2 or 4 judges Assign an admissible currently non assigned judge if possible Otherwise eliminate 1 judge (making sure that the competition has a lead judge assigned) New initial solution x 0 is used as a new initial solution for the next major iteration

20 Numerical Results 4 variants: H-Best, R-Best, H-First, R-First Initial solution : Heuristic (H), Random (R) Selection strategy: Best, First 3 sets of randomly generated problems P 1, P 2, P 3 In each set: subsets (10 problems) with 15, 50, 150, and 500 competitions P 1 : some comp. with 3 judges; judges with diff. expertise in all comp. P 2 : all comp. with 5 judges; in some comp.,judges with same expertise P 3 : some comp. with 3 judges; in some comp.,judges with same expertise

21 Worth using metaheuristic CPLEX: much more CPU Variants: very small Ave dev Ave dev < 1 At most one competition where 2 judges have the same expertise Robustness Variants: solutions of excellent quality for all problems

22 Judge Asssignment for the 5 rounds A judge cannot be assigned to a competition involving a team that he does not wish to evaluate If a team in a competition is presenting in French, then the Constraints judges assigned to this competition must be fluent in French At least one experienced judge, different from the lead judge, must be assigned to each competition The judges assigned to each competition should be balanced with regard to the number of experienced and new judges Objective function If several judges coming from firms are assigned to a competition then they should come from different ones o Same pair of judges cannot be assigned more than once o Same judge cannot be assigned more than once at different competitions of any given team

23 Solution procedure including two phases: Phase 1: Initialisation to generate an initial solution Phase 2: Iterative process to improve the current solution Phase 1 ( p = ) For each round p 1,,5 considered individually, determine an initial solution xinit accounting for the pairs of judges already p assigned for the other rounds already completed 5 U xbest : = xinit the best solution generated so far p= 1 5 U xlast : = xinit the best current solution p= 1 p p

24 Phase 2 (Iterative process repeated until a stopping criterion is met) At each iteration of the process: Generate a permutation P of the the rounds where some constraints are violated For each round p P, o Fix the assignments of the judges in all the other rounds o * Determine a new optimal solution xp for round ( ) * Use the earlier procedure using o xlast : = xlast xlast p U x Another procedure also using p tabu searches and diversification tabu search and diversification o If adapted xlastto is account better for than the xbest, then xbest : = xlast have also been developed. additional Interpretation constraints underlying an iteration: Gauss-Southwell type approach to Having the global problem in view, solve the global 5 rounds problem solving the round p sub-problem is allowing to modify only the assignments equivalent to consider neighborhood for each round sequentially for the global problem where only assignments of round p can be modi fied. Variable neighborhood type procedure p

25 Numerical Results 2 sets of randomly generated problems P 1, P 2 P 1 : some competitions with 3 judges P 2 : all competitions with 5 judges In each set: subsets (10 problems) with 15, 30, and 90 competitions per round Worth using metaheuristic CPLEX: fails to find an integer feasible solution in 10 hours Variants: solutions of good quality in less than 10 seconds

26 Software Solution approach embedded into a user friendly software Results on real data to the full satisfaction of organizing committee.

27 CPLEX

28 Worth using metaheuristic CPLEX much more CPU Heuristic very small Ave dev

29 Ave dev < 1 At most one competition where 2 judges have the same expertise

30 Robustness H-Best, H-First, R-First: Optimal value or lower bound achieved for at least one solution out of 5 R-Best: Also verified except for 2 instances of P 3 with 150 comp., and 1 of P 3 with 500 comp.

31 f Heuristic initial solution is better H-Best dominates H-First dominates R-Best R-First

32

33 H-First vs H-Best H-First: better solutions

34 H-First vs H-Best H-First: better solutions H-Best: smaller CPU

35 H-First vs H-Best Problem of size 150 H-First: better solutions factor 7 H-Best: smaller CPU factor 2

36 H-First vs H-Best Problem of size 150 H-First: better solutions factor 7 H-Best: smaller CPU factor 2 Problem of size 500 H-First: better solutions factor 2 H-Best: smaller CPU factor 4

37 H-First vs H-Best H-First: better solutions H-Best: smaller CPU factor increasing with problem size

38 R-First vs R-Best Similar results

39 Conclusion All variants generate solutions of excellent quality With regards to CPU: H-Best is slightly dominating